Line two rimmed baking sheets with parchment paper. Zest two large oranges before juicing them - you'll need the zest for both the dough and glaze.

In a large bowl with an electric mixer, beat softened butter and granulated sugar on medium speed until light and fluffy, about 3 minutes. Scrape down the bowl once or twice.

Beat in the egg until fully incorporated, then mix in 2 tablespoons orange zest and 2 tablespoons fresh orange juice. The mixture will look slightly curdled - that's fine.

Add flour, baking powder, and salt. Mix on low speed just until no dry streaks remain and a soft dough forms. Do not overmix.

Scoop heaping tablespoonfuls of dough and roll into balls. Place 2 inches apart on prepared baking sheets. Flatten each ball slightly with the bottom of a glass to about 1/2 inch thick. Chill in the refrigerator for 15 minutes while you preheat the oven to 350°F.

Bake until the edges are just set and the tops look matte with small cracks, 10 to 12 minutes. The centers should still be soft. Cool on the baking sheets for 5 minutes, then transfer to wire racks to cool completely.

Whisk powdered sugar and 2 tablespoons orange juice until smooth and pourable. Spoon about 1 teaspoon glaze onto each cooled cookie, letting it drip toward the edges. Sprinkle with remaining 1/2 teaspoon zest while wet. Let set for 20 minutes before serving.